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Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

Giorgos Arampatzis, Markos A. Katsoulakis, Petr Plechac, Michela Taufer, Lifan Xu
Department of Applied Mathematics, University of Crete and Foundation of Research and Technology-Hellas, Greece
arXiv:1105.4673v1 [math.NA] (24 May 2011)

@article{2011arXiv1105.4673A,

   author={Arampatzis}, G. and {Katsoulakis}, M.~A. and {Plechac}, P. and {Taufer}, M. and {Xu}, L.},

   title={"{Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1105.4673},

   primaryClass={"math.NA"},

   keywords={Mathematics – Numerical Analysis, Physics – Computational Physics},

   year={2011},

   month={may},

   adsurl={http://adsabs.harvard.edu/abs/2011arXiv1105.4673A},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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We present a mathematical framework for constructing and analyzing parallel algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. The algorithms can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms are controlled-error approximations of kinetic Monte Carlo algorithms, departing from the predominant paradigm of creating parallel KMC algorithms with exactly the same master equation as the serial one. Our methodology relies on a spatial decomposition of the Markov operator underlying the KMC algorithm into a hierarchy of operators corresponding to the processors’ structure in the parallel architecture. Based on this operator decomposition, we formulate Fractional Step Approximation schemes by employing the Trotter Theorem and its random variants; these schemes, (a) determine the communication schedule} between processors, and (b) are run independently on each processor through a serial KMC simulation, called a kernel, on each fractional step time-window. Furthermore, the proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms, showing the convergence of our approximating schemes to the original serial KMC. The approach also provides a systematic evaluation of different processor communicating schedules.
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